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32
THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS
, 2011
"... Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new univ ..."
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Cited by 97 (15 self)
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Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object – a non-linear stochastic partial differential equation – known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
The effect of disorder on polymer depinning transitions
- Commun. Math. Phys
"... Abstract. We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by n −c ϕ(n) for some 1 < c < ..."
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Cited by 49 (4 self)
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Abstract. We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by n −c ϕ(n) for some 1 < c < 2 and slowly varying ϕ. Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u) to the corresponding annealed system. We show that for c> 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as β 1/(2c−3) where β is the inverse temperature. For c < 3/2, given ǫ> 0, for sufficiently high temperature the quenched and annealed curves are within a factor of 1 − ǫ for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function ϕ. 1.
A replica-coupling approach to disordered pinning models
- Comm. Math. Phys
"... Abstract. We consider a renewal process τ = {τ0, τ1,...} on the integers, where the law of τi − τi−1 has a power-like tail P(τi − τi−1 = n) ≃ n −(α+1) with α ≥ 0. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this ..."
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Cited by 37 (4 self)
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Abstract. We consider a renewal process τ = {τ0, τ1,...} on the integers, where the law of τi − τi−1 has a power-like tail P(τi − τi−1 = n) ≃ n −(α+1) with α ≥ 0. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this class of problems includes, among others, (1 + d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1 + 1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase where τ occupies a finite fraction of N to a delocalized phase where the density of τ vanishes. In absence of disorder (i.e., if the reward is n-independent), the transition is of first order for α> 1 and of higher order for α < 1. Moreover, for α ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as α> 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for α < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, applying techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2].
Fractional moment bounds and disorder relevance for pinning models
- COMMUN. MATH. PHYS
, 2007
"... We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)-locali ..."
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Cited by 36 (16 self)
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We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [2, 24]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [13, 9], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α> 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed
Quenched and annealed critical points in polymer pinning models
"... Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spe ..."
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Cited by 34 (6 self)
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Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form n −c ϕ(n) with c ≥ 1 and ϕ slowly varying. Comparing to the corresponding annealed system, in which the Vn are effectively replaced by a constant, it was shown in [1], [4], [11] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c> 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided ϕ(n) → 0 as n → ∞, and for c> 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2. 1.
DISORDERED PINNING MODELS AND COPOLYMERS: BEYOND Annealed Bounds
, 2008
"... We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the Poland-Scheraga model of DNA denaturati ..."
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Cited by 32 (4 self)
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We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the Poland-Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of non-integer moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, e.g., if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0 < α < 1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a âreduced wetting model â introduced by T. Bodineau and G. Giacomin [5]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
- Ann. Probab
"... Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and w ..."
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Cited by 25 (4 self)
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Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x–axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay non-negative. We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε> ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are non-trivial and distinct: 0 < ε p c < ε w c < ∞, and they are the only non-analyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field. 1. Introduction and
EQUALITY OF CRITICAL POINTS FOR POLYMER DEPINNING TRANSITIONS WITH LOOP EXPONENT ONE
, 811
"... Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned ..."
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Cited by 18 (2 self)
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Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length n takes the form ϕ(n)/n for some slowly varying ϕ; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. 1.
A renewal theory approach to periodic copolymers with adsorption
, 2007
"... We consider a general model of a heterogeneous polymer chain fluctuating in the proximity of an interface between two selective solvents. The heterogeneous character of the model comes from the fact that the monomer units interact with the solvents and with the interface according to some charges th ..."
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Cited by 12 (7 self)
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We consider a general model of a heterogeneous polymer chain fluctuating in the proximity of an interface between two selective solvents. The heterogeneous character of the model comes from the fact that the monomer units interact with the solvents and with the interface according to some charges that they carry. The charges repeat themselves along the chain in a periodic fashion. The main question concerning this model is whether the polymer remains tightly close to the interface, a phenomenon called localization, or whether there is a marked preference for one of the two solvents, thus yielding a delocalization phenomenon. In this paper, we present an approach that yields sharp estimates for the partition function of the model in all regimes (localized, delocalized and critical). This, in turn, makes possible a precise pathwise description of the polymer measure, obtaining the full scaling limits of the model. A key point is the closeness of the polymer measure to suitable Markov renewal processes, Markov renewal theory being one of the central mathematical tools of our analysis.
